Extremely confined gap plasmon modes: when nonlocality matters

Historically, the field of plasmonics has been relying on the framework of classical electrodynamics, with the local-response approximation of material response being applied even when dealing with nanoscale metallic structures. However, when the confinement of electromagnetic radiation approaches atomic scales, mesoscopic effects are anticipated to become observable, e.g., those associated with the nonlocal electrodynamic surface response of the electron gas. Here, we investigate nonlocal effects in propagating gap surface plasmon modes in ultrathin metal–dielectric–metal planar waveguides, exploiting monocrystalline gold flakes separated by atomic-layer-deposited aluminum oxide. We use scanning near-field optical microscopy to directly access the near-field of such confined gap plasmon modes and measure their dispersion relation via their complex-valued propagation constants. We compare our experimental findings with the predictions of the generalized nonlocal optical response theory to unveil signatures of nonlocal damping, which becomes appreciable for few-nanometer-sized dielectric gaps.


PDMS
. a Synthesis of gold flakes on BK-7 glass substrate; b Coating sample with ∼ 6 nm conductive carbon (C) layer for subsequent SEM and FIB milling; c Identification of a suitable gold flake sample on the substrate using optical and scanning electron microscopy, with subsequent milling of the coupling-element slits; d Removal of carbon layer and atomic-layer deposition (ALD) of aluminum oxide (Al 2 O 3 ) layer; e Transfer of a thin gold flake from another substrate onto the milled structure using a polydimethylsiloxane (PDMS) stamp; f Top view and g cross-section of the final structure.

S4. ANALYSIS OF THE NEAR-FIELD DATA
Here we describe the procedure of processing measured near-field data. For all samples the scanned length and width was 10 and 3 microns, respectively, with the waveguide coupler being in the center. Although the top gold flake covers the waveguide coupler, hindering its easy identification via topography (Fig. S5a), it was possible to identify its position by two distinct lines in near-field amplitude (Fig. S5b), corresponding to the position of the major slit in the waveguide coupler (see inset in Fig. S5a). Next, we selected area outside of the coupler (green dashed line in Fig. 5b) and applied extended discrete Fourier transform (EDFT) [S1, S2] along the x-axis to identify propagating waves (Fig. S5d,e). The GSP mode appears as a pronounced peak with an effective mode index k x /k 0 > 2 (around 5 for the alumina thickness of 3 nm, Fig. S5e). Then we filtered this mode with a reasonably wide rectangular apodization function (with a width of 3.5 for 2 nm alumina; 2.5 for 3 and 5 nm alumina; and 2.2 for 10 and 20 nm alumina). The result of Fourier filtering, E FT (Fig. S5f,g) was then line-by-line fitted with a complex exponent function in order to precisely determine the real and imaginary part of n GSP (Fig. S5h). The fitted field distribution, E fit , and residuals can be found in Fig. S5i-j and k-l, correspondingly. The goodness of fitting for each line was determined as 1 − |E FT −E fit | 2 dx |E FT | 2 dx , reaching 1 for the perfect fit and 0 for no fit at all. For each alumina thickness there were 5-10 scans, done in different positions of the sample, resulting in 150-300 points with n GSP . The weighted mean was then used to determine the average n GSP , with weighted standard deviation used to determine the error bars for each alumina thickness, with the goodness of fitting used as a weight. d Amplitude of the Fourier transform of the recorded near-field along x-axis and e its y-axis average. Red line in e depict used rectangular apodization function, used to filter GSP mode. f, g Amplitude and phase of the near-field after Fourier filtering. h GSP effective mode index for each y-axis line, acquired by fitting the filtered field with a complex exponent. i-l Fitted field and residuals.

S5. FITTING THE DIFFUSION CONSTANT D
Estimation of the diffusion constant D was performed by fitting the GNOR dispersion relation (eq. 1a) (with D being a free parameter) to the imaginary part of the experimentally obtained effective mode index n GSP for the corresponding experimental values of real part of n GSP . Since Re{n GSP } is nearly independent of D (see Fig. S4), this permits to make a fit to solely imaginary part, by considering only those thicknesses t d which satisfy the experimental value of Re{n GSP }. Fig. S6 compares the result of our fit against the values reported in previous works, summarized in Table S1.    [S8-S11] in comparison with experimentally-measured data points. a Parametric plot of the dispersion curves; b real and c imaginary part of the effective mode index versus nominal dielectric gap thickness.

S8. INFLUENCE OF AN ADDITIONAL AIR VOID
One of the possible explanations for persistently smaller real part of the experimentally retrieved n GSP (see Fig. S4b) could be existence of an additional air layer between dielectric ALD layer and upper gold flake. As mentioned in the main text, presence of such air void could be caused by imperfections in the sample fabrication, for example contamination during the transfer step or inhomogeneous adhesion of the upper gold flake to the ALD layer. This results in local variation of the gap thickness and effective permittivity of the dielectric layer, consequently broadening peak corresponding to n GSP in the real part of k x spectrum (see Fig. S3d). Here, we demonstrate implications of such a thin air layer for the dispersion relation and show compatibility of this assumption with the interpretation of our experimental results.
We suppose that the gap between metal flakes is composed of alumina layer (of nominal thickness t ALD calculated by multiplying the number of ALD cycles by the average ALD alumina growth rate) and an additional layer of air (with thickness t air ) , see Fig. S10. This results in a lower effective dielectric constant of the material in the gap, which is calculated using the effective medium theory, according to the formula: with f air = t air t air +t ALD and f ALD = t ALD t air +t ALD being air and alumina volume factors, correspondingly. An additional air void results not only in reduced effective dielectric constant of the gap, but also in increased total gap size, which both leads to the reduced real and imaginary parts of the GSP effective mode index (as can be seen in Fig. S11, for both local and nonlocal models).However, when dispersion is plotted in the parametric graph (Fig. S11a), the addition of the air layer has negligible influence on the curve position, similarly to the negligible influence of dielectric constants of the material in the gap [ Fig. S8a]. This allows to experimentally quantify the influence of non-local corrections without precise knowledge of gap composition. The real part of the effective mode index n GSP depends on both effective permittivity of the gap material ε d,eff and gap thickness t d . Since non-local corrections have negligible influence on the real part of n GSP for our range of gaps (Fig. S11), one can find pairs of ε d,eff and t d , resulting in the same Re{n GSP } as was experimentally measured (black solid lines in Fig. S12, with gray margins corresponding to the experimentally measured error bar of Re{n GSP }). Next, assuming the gap composition as ALD alumina layer of thickness t ALD with additional air layer of thickness t air (see previous section), such that t d = t air + t ALD , one can plot effective permittivity of the gap, ε d,eff vs. air layer thickness, t air (dashed colored lines in Fig. S12). The intersection of these curves with previously defined black solid lines provides estimation of the additional air layer thickness in measured samples, which was found to be (see table I).    Table S2. Optimized geometric parameters of the tapered couplers for the MDM waveguides with various thicknesses of dielectric gap t d .

Modified boundary conditions
Feibelman d-parameters can be incorporated into the conventional boundary conditions, by considering surface (normal) polarization P(r) = π(r)δ(r − r δΩ ) and surface (tangential) current J(r) = K(r)δ(r − r δΩ ) which are driven by the discontinuities of the E and D fields [S12]: